INVESTIGATION ON THE MODEL OF VORTEX-INDUCED

Transcription

1 The Seventh Asi-Pcific Conference on Wind Engineering, November 8-1, 9, Tipei, Tiwn ABSTRACT INVESTIGATION ON THE MODEL OF VORTEX-INDUCED VIBRATIONS OF RECTANGULAR SUPER HIGH-RISE BUILDINGS Hi-Yng Wu 1 Shu-Guo Ling Zheng-Qing Chen 3 Shu-Ling Wng 4 1 Engineer, Centrl Southern Chin Electric Power Design Institute, Wuhn 4371,Chin,wuocen1981@163.com Professor, School of Civil & Building Eng., Wuhn Univ., Wuhn 437, Chin, lingsg@public.wh.hb.cn 3 Professor, Wind Engineering Reserch Center, Hunn Univ., Hunn 418, Chin, zqchen@hnu.cn 4 Post grdute, School of Civil & Building Eng., Wuhn Univ., Wuhn 437, Chin,div1985@16.com Bsed on the empiricl non-linel model, Vickery-Bsu model nd generlized Vn der Pol oscilltor model, two improved generlized Vn der Pol oscilltor modes. i.e. IGVPO-1 nd IGVPO- re proposed for evluting dynmic responses of rectngulr high-rise buildings when lock-in occurs. IGVPO-1is suitble for predicting the mplitude of displcement response t ny lock-in velocity, nd IGVPO- is suitble for predicting the mximum mplitude of displcement response when lock-in occurs. Finlly, the erodynmic prmeters in the two models re identified bsed on the ero-elstic model wind tunnel test dt. Compring the wind tunnel dt with those predicted by the two models, the results indicted tht the two models estblished in this pper re of much high precision. KEYWORDS: RECTANGULAR HIGH-RISE BUILDING, AERO-ELASTIC MODEL WIND TUNNEL TEST, VORTEX-INDUCED VIBRATION Introduction As the heights of high-rise buildings increse, which induces the decrese of the nturl frequencies of the buildings, the risk of vortex-induce vibrtions of the super high-rise buildings under strong wind becomes greter. Vortex-induced vibrtions of super high-rise buildings cn be divided into two sttes: one is hrmonic vibrtion tht hppens when lock-in occurs, nd the other is nrrow-bnd rndom vibrtion. This pper minly discusses the clcultion models nd nlysis methods of the first kind of vortex-induced vibrtion. Investigtion on the models of vortex-induced vibrtion ws strted in 196s. The existing models include lift-oscilltor model [Skop et l.(1975)], empiricl linel model [Scnln (1976) nd Gde et l. (1976)], empiricl non-linel model [Simu et l. (1996)], correltion model [Blevins (1983)], generlized vn der Pol oscilltor model [Lrsen (1995)] nd Vickery-Bsu model [Vickery et l. (1983)] nd etc.. Through nlyzing, it is shown tht lift-oscilltor model, empiricl linel model, empiricl non-linel model, correltion model nd generlized vn der pol oscilltor model re ll suitble for the D flow field. Although encircled effect of 3D flow is considered in Vickery-Bsu model, it is just suitble for

2 structures with circulr sections. In this pper, two improved generlized Vn der Pol oscilltor models, i.e IGVPO-1 nd IGVPO-, re proposed on the bse of empiricl non-linel model, Vickery-Bsu model nd generlized Vn der Pol oscilltor model, nd proved to be effective by wind tunnel tests of ero-elstic tll building model. Estblishment of the Lock-in Model of Rectngulr Super High-rise Buildings Bsic Hypothesizes Aerodynmic forces, especilly those in cross wind direction, cting on the super high-rise buildings under strong wind re inherently complex. For convenience, it is generlly ssumed tht cross-wind erodynmic forces result from the liner superposition of two prts [Simu et l. (1996) nd Cheng et l.(1996) nd Chen et l. (1997)]. The first prt is the cross-wind erodynmic lods on the sttionry building; the other prt is body motion induced force. It is proved tht when lock-in occurs, the first prt is insignificnt compred with the second one[simu et l. (1996) nd Ehsn et l.(199)]. The First Improved Generlized Vn der Pol Oscilltor model (IGVPO-1) Bsed on the wind tunnel dt nd tking ccount of lock-in rnge of reduced wind velocities, the first improved generlized Vn der Pol oscilltor model (IGVPO-1) is proposed in this pper. This model cn be used to predict the mplitude of displcement response t ny reduced wind velocity in lock-in rnge. For convenience, the single DOF system is considered, nd IGVOP-1 cn be expressed s: y m( y+ ξω s y + ωy) = ωρ D K 1 ε y D (1) where, m is mss, ξs is structurl dmping rtio, ω is the vibrtion frequency considering the dded mss of ir round the vibrting body, y nd y re the first order derivtive nd second order derivtive of time respectively, ρ is the density of the ir, D is the width of windwrd surfce. k = V () V cr Where V is commonly known reduced velocity, V cr is the criticl reduced velocity t lock-in strt, K, ε nd v re prmeters which must be identified by wind tunnel dt, nd re the function of k. The non-dimensionl displcements cn be expressed in terms of y η = (3) D And let η = η cosω t Finlly, the mplitude of displcement response corresponding to certin wind velocity cn be deduced from IGVPO-1, (4)

3 where In which 1 π Sc η = 1 Ic()() v k C ( k ) (5) ε () = π v cos( p) sin ( p) Ic d (6) Mξ s Sc = ρ D Mξ C = ρd 1 ε η M φ ( z) H = H ( ) ϕ ( ) m z ϕ ( ) z dz z dz is vibrtion mode, H is height of building, p ξ is erodynmic dmping rtio. When v =1, the right side of eqution (1)is non-linel self-excited force bsed on vn der pol π π oscilltor model. In this cse, Ic = cos ( p) sin ( p) dp =, consequently: Mξ = 1 s η (1) ε ρ D K Y1 If K =, in which S is Strouhl number, Y1 stnds for erodynmic prmeter, substituting 4π S them into eqution(1) led to: η = 1 16πξsMS 4ρD Y1 ( K ) ρy1 ( K) εd (11) tht is sme s the clculting formul of the vortex-induced resonnce vibrtion mplitude deduced from empiricl non-linel model. The Second Improved Generlized Vn der Pol Oscilltor Model (IGVPO-) The structure designers re more interested in the mximum vortex-induced vibrtion mplitude. The second generlized Vn der Pol oscilltor model could stisfy this requirement. It is proposed to predict the mximum mplitude of displcement in lock-in rnge, which cn be expressed s ε y my ( + ξω s y + ωy) = ωρ DK 1 y γ (1) ( kmx ) D where, Vmx k = (13) mx V cr where V is the mximum reduced velocity in the rnge of lock-in, K, ε, v ndγ re mx prmeters need to be identified by experiment dt,nd no longer the functions of k. (7) (8) (9)

4 By the sme method, the mximum mplitude of displcement response cn be obtined η ( kmx ) ( ) = Ic v ε γ π mx 1 ( Sc C ) 1 (14) Wind Tunnel Test nd Identifiction of Aerodynmic Prmeters Considering schemtic digrm of the experimentl setup shown in Fig.1, the building model is rigid model with connecting rod jointed t the bottom. It is instlled on the wind tunnel floor with the model bse pivoted. When the model swys with liner mode shpe, two couples of springs below the floor supply orthogonl horizontl return forces, nd oil in tnk under the model djusts dmping of the model. The spect rtio nd side rtio of the rigid model is 9:1 nd 1:1 respectively. The wind tunnel tests were crried out in HD- wind tunnel, Hunn University, Chngsh, Chin. Only norml ttck ngle nd smooth wind flow were considered. The test cses re shown in Tble1. Figure1: Schemtic Digrm of Experimentl Model nd Setup Tble1 Test cses cses frequency H B D ξ s (%) Mξ s (Hz) M H BD Sc = (cm) (kg/m) ρ D X Y X Y The lock-in rnge chnging with Sc The cross-wind root-men-squre displcements of the rectngulr building models with different Sc chnged long with the reduced wind speed in the uniform wind field is shown in Fig.. It cn be seen tht when the reduced wind speed V reches bout 1.6, f D the vibrtion mplitudes of ll models increse rpidly. This reduced wind speed is the criticl y

5 reduced velocity of vortex-induced resonnce, i.e.. After lock-in strts, it tkes plce t certin rnge of wind velocities. It is lso cn be found tht the vibrtion mplitude decreses nd lock-in rnge becomes nrrow with the increse of Sc number.. Let indicte the mximum reduced wind velocity of lock-in rnge, defining k V mx V V mx mx = nd extrct V mx from Fig. to clculte mx, so tht the chnging lw of cr with Sc number cn be deduced s shown by Eq.(15 ) nd in Fig. 3. k mx 1 ( Sc Sc ) t1 ( Sc Sc ) t V cr = k + A e + A e (15) where k = 1.44, Sc =.35, A 1 = 1.83, t 1 = 1.66, A = 7.745, t =.358. k k mx Figure Across-wind RMS Displcements t Top of the Models with Different Sc number versus reduced velocity Figure3 Chnging lw of k mx long with Sc number Identifiction of erodynmic prmeters of IGVPO-1 Becuse of the limittion of wind tunnel dt, the erodynmic prmeters cquired in this pper only suitble for1 < k < In order to predict lock-in responses mplitudes of models with smller Sc number under strong wind, more wind tunnel dt is needed, but the method is the sme s below. For ny wind velocity in lock-in rnge, vortex-induced vibrtion mplitude cn be predicted by IGVPO-1, which is given s: 1 π Sc η = 1 Ic()() v k C ( k) (16) ε

6 where: () = π v cos( p) sin ( p) Ic d (16) p By regression nlysis on the bsis of wind tunnel dt, we obtin ( k) ( 1 < k < 1.5) k ( 1.5 k 1.6) ( k) ( 1.6 < k 1.7) k ( 1.7 < k 1.76) ε ( k) = (16b) k k ( 1 < k < 1.5) k 33655k k ( 1.5 k 1.6) C ( k) = (16c) k ( 1.6 < k 1.7) k ( 1.7 < k 1.76) 1.37 ( k) ( 1 < k < 1.5) 1 ( ) ( 1.5 k 1.6) v k = (16d) ( k ) ( 1.6 < k 1.7) k ( 1.7 < k 1.76) It demonstrtes much precision of formul (16) by compring the cross-wind displcement mplitude of the rectngulr high-rise building models with different Sc number with the theoreticl predicted vlues, s shown in Fig. 4. () k = 1.1 (b) k = 1.3 (c) k = 1.5 (d) k = 1.7 Figure 4 Comprisons of predicted vlues of IGVPO-1 with test vlues under different wind velocities

7 Identifiction of erodynmic prmeters of IGVPO- IGVPO- cn be used to predict the mximum displcement mplitude when lock-in occurs, nd the predicting formul nd erodynmic prmeters re given s below: η ( kmx ) Ic() v γ π = 1 ε ( Sc ) C 1 whereε = ; C = 1; v =. 45; γ =.959 Fig.5 shows comprison between the mximum displcement mplitudes during lock-in in rnge of Sc numbers, which obtined from test dt nd those predicted by different theoreticl models respectively. (17) Figure 5 The mximum displcement mplitudes during lock-in versus Sc The results show good greement between test dt nd IGVPO- curve. As similr to the results obtined by Lrsen (1995), Empiricl non-linel model is bsed on the vn der pol oscilltor model (VPO), nd the predicted vlues of VPO cn not conform to the test vlues well, even the shpe of curves is different. Downtrend of the predicting curve of GVPO conforms to the test vlues, but there re still some differences in numericl vlues, which is not s perfect s the fitting of GVPO curve versus experimentl dt for D cylinder presented by Lrsen(1995). In contrst, the displcement curve evluted by IGVPO- cn be dpted to experimentl dt very well. Conclusion The gretest dvntge of the improved generlized Vn der pol oscilltor models proposed in this pper is tht it hs tken ccount of the rnge of lock-in, nd could predict the mximum displcement mplitude nd the displcement mplitudes of rectngulr tll buildings under ny wind speed of lock-in rnge. The improved generlized Vn der pol oscilltor models proposed in this pper provide prcticl pproch to investigte dynmic responses of non-circulr super high-rise buildings, nd lso estblish two theoreticl models of vortex-induced vibrtion, which become useful references for further investigtions on vertex-induced lock-in of non-circulr supper high-rise buildings. In the men time, the eqution to predict the lock in rnge of wind speeds for high rise buildings with different Sc number is lso proposed on the bsis of wind tunnel dt in this pper. Though IGVPO-1 nd IGVPO- re deduced bsed on uniform flow field test dt, they re lso pplicble to turbulent flow field. In turbulent flow field, lock-in my tke plce for tll buildings with very smll Sc numbers[wu(8), Kwok et l.(1981) nd Ling(1996)], so tht the erodynmic prmeters in theoretic models cn lso be identified from lock-in test

8 dt of turbulent flow field. As the heights of high-rise buildings increse, which induces the decrese of the nturl frequencies of the buildings, the risk of vortex-induce vibrtions of the super high-rise buildings under strong wind becomes greter. Consequently, to reserch the method of predicting the vortex-induced vibrtion of high-rise buildings in turbulent flow field on the bsis of this pper becomes extremely necessry. Acknowledgements The reserch presented in this pper ws supported by the Chin Ntionl Science Foundtion under project No , which is grtefully cknowledged. References Skop R.A. nd Griffin O.M. (1975), On Theory for the Vortex-Excited Oscilltions of Flexible Cylindricl Structures, Journl of Sound Vibrtion, 41: Scnln R.H.(1975), Theory of the Wind Anlysis of Long-Spn Bridges Bsed on Dt Obtin-ble from Section Model Tests, Proceedings of the Fourth Interntionl Conference on Wind Effects, London, Cmbridge Univ. Press. Cmbridge, U.K.: Simiu, E nd Scnln, R. H. (1996), Wind Effects on Structures, 3 rd Edition, John Wiley & Sons. Ehsn F., Scnln R. H. (199), Vortex-Induced Vibrtions of Flexible Bridges, Journl of Engineering Mechnics, Blevins R.D.(1983). Flow induces vibrtion. Wu Shusn, Wng Jue trnslted. Beijing:Mchinery Industry Press.(in Chinese) Lrsen A. (1995), A Generlized Model for Assessment of Vortex-induced Vibrtion for Flexible Structures Journl of Wind Engineering nd Industril Aerodynmics, 57: Vickery B.J. nd Bsu R.I.(1983), Across-Wind Vibrtions of Structures of Circulr Cross-Section, Prt I: Development of Two-Dimensionl Model for Two-Dimensionl Conditions, Journl of Wind Engineering nd Industril Aerodynmics, 1: Cheng C. M., Chen R. H., Wen B. J., et l. (1996), Influence of Across-wind Vibrtion to the Wind Chrcteristics of wind lods of Rectngulr Columns, Republic of Chin, 8th Ntionl Mechnics Conference, Tipei, Decembe, (in Chinese) Chen R. H. Cheng C.M, Wen B. J. (1997), Aerodynmic Interction Phenomenon Between High-rise Buildings nd the Boundry Lyer Flow Field, Journl of Chin Civil nd Hydrulic Engineering, 9(): (in Chinese) Wu Hiyng (8), Investigtion on Vortex-inudced Vibrtion of Rectngulr Super High-rise Buildings by Wind Tunnel Test, PH.D. Thesis, Wuhn University, Wuhn, Chin. (in Chinese) Kwok K. C. S. nd Melbourne W.H. (1981), Wind-induced lock-in excittion of tll structures, Journl of the Strcturl Division, ASCE.17(ST1): Ling S.G. (1996), Reserch on the Vortex-induced Vibrtion nd Glloping of Tll Buildings nd Structures, The project report of the stte key lbortory of civil engineering disster prevention of Tongji university. (in Chinese)